Continuously improving the accuracy and precision of planning and scheduling models is not new; unfortunately it is not institutionalized in practice. The intent of this paper is to highlight a relatively simple approach to historize or memorize past and present actual planning and scheduling data collected into what we call the past rolling horizon (PRH). The PRH is identical to the future rolling horizon (FRH) used in hierarchical production planning and model predictive control to manage omnipresent uncertainty in the model and data. Instead of optimizing future decisions such as throughputs, operating-modes and conditions we now optimize or estimate key model parameters. Although bias-updating using a single time-sample of data is common practice in advanced process control and optimization to incorporate “parameter” feedback, this is only realizable for real-time applications with comprehensive measurement systems. Proposed in this paper is the use of multiple synchronous or asynchronous time-samples in the past in conjunction with simultaneous reconciliation and regression to update a subset of the model parameters on a past rolling horizon basis to improve the performance of planning and scheduling models.

1. CONTINUOUSLY IMPROVE THE PERFORMANCE
OF PLANNING AND SCHEDULING MODELS WITH
PARAMETER FEEDBACK
Jeffrey D. Kelly* and Danielle Zyngier
Abstract
Continuously improving the accuracy and precision of planning and scheduling models is not new;
unfortunately it is not institutionalized in practice. The intent of this paper is to highlight a relatively
simple approach to historize or memorize past and present actual planning and scheduling data collected
into what we call the past rolling horizon (PRH). The PRH is identical to the future rolling horizon
(FRH) used in hierarchical production planning and model predictive control to manage omnipresent
uncertainty in the model and data. Instead of optimizing future decisions such as throughputs, operating-
modes and conditions we now optimize or estimate key model parameters. Although bias-updating using
a single time-sample of data is common practice in advanced process control and optimization to
incorporate “parameter” feedback, this is only realizable for real-time applications with comprehensive
measurement systems. Proposed in this paper is the use of multiple synchronous or asynchronous time-
samples in the past in conjunction with simultaneous reconciliation and regression to update a subset of
the model parameters on a past rolling horizon basis to improve the performance of planning and
scheduling models.
Keywords
Rolling horizons, reconciliation, parameter estimation, error-in-variables, closed-loop, feedback.
Introduction
* To whom all correspondence should be addressed. jdkelly@industrialgorithms.ca (Industrial Algorithms LLC.)
Planning and scheduling decision-making is
traditionally based on simplified models that can, with any
luck, accurately and precisely interpolate and extrapolate
the dominate behavior of the underlying production in
terms of its processes, operations and maintenance.
Related to this is the well known notion that "all models
are wrong, but some are useful" (G.E.P. Box) which
implies that even detailed models do not guarantee their
accuracy and precision in being able to predict and
optimize production (Forbes and Marlin, 1994, Zyngier
and Marlin, 2006).
The focus of this paper is to suggest a symmetrical
methodology to the future rolling horizon (FRH, Baker and
Peterson, 1979) we call the past rolling horizon (PRH)
which implements a continuous improvement strategy
similar to the Deming Wheel, Shewhart Cycle, Kaizen or
the Plan-Perform-Perfect-Loop (Kelly, 2005). The use of a
past rolling horizon for planning and scheduling is a
similar concept to moving-window estimators (Robertson
et al., 1996, Zyngier et al., 2001, Yip and Marlin, 2002). In
the PRH, “parameter” optimization is performed with
essentially fixed variables looking backwards in time
whereas in the FRH of planning and scheduling we use
“variable” optimization with fixed parameters looking
forwards in time where fixed implies exogenously defined
as opposed to endogenously determined via the
optimization process.

2. The structure of paper is to first illuminate the
different aspects of a model, second to highlight the issues
with typically passive data collection, third, updating and
estimating techniques are discussed and fourth a
motivating example is overviewed to demonstrate the need
for what we call “parameter” feedback and not just
“variable” feedback used in existing planning and
scheduling implementations.
Model Morphology
For decision-making found in the process industries,
models can usually be segregated into three different types.
(1) Macro or lumped models are the ones that usually
consider an entire plant (or a large section thereof) taking
into account only the core or critical unit-operations as
well as aggregations of units. Macro models are commonly
encountered in planning, scheduling and yield accounting
applications; (2) Micro or distributed models are more
restricted in scope than macro models in that they usually
consider a smaller number of unit-operations and are
usually distributed across several spatial dimensions
including time. The modeling of these unit-operations
comprise more detailed mass, energy and momentum
balances including vapor-liquid equilibrium and reaction
kinetic relationships. Micro models are widely applied in
advanced process simulation and optimization; and finally
(3) Molecular models address the relationships that occur
at an atomic or elemental level of granularity within a
small section of a unit-operation with very detailed
thermodynamics and transport phenomena.
Whether a model is macro, micro or molecular, there
are three relevant aspects of the model morphology. A
model may be classified by its structural form such as if it
is linear, piece-wise linear, polynomial, rational, multi-
linear or non-linear. On the other hand, its functional form
relates to its parameters, coefficients and/or factors. Its
syntactical form relates to how the model functions,
formulae or formulations are expressed. Syntactically,
models can be explicit or implicit (i.e., use closed- or open-
form1 equations respectively) of which the latter is the
more general form i.e., comprises explicit models as a
subset.
Other aspects of a model such as whether it is static or
dynamic (steady or unsteady), continuous or discrete and
deterministic or non-deterministic (stochastic or chaotic) is
also worth mentioning. Planning and scheduling models
are usually dynamic in the sense of having multiple time-
periods built-up from essentially static models, have a mix
of continuous and discrete variables to represent the
quantity, logic and quality dimensions and are mostly
deterministic. Another important aspect of a model is
related to its fidelity and size. Bigger and more detailed
models do not guarantee its precision or accuracy as shown
1 There can also be “pried-open” models which break-apart the
internal convergence loops inside closed-form models.
by Forbes and Marlin (1994), i.e., smaller and simpler
models can be just as “useful” if they meet certain point-
wise model accuracy criteria. Therefore, we can class
models into being either rigorous or rough. Rough models
are related to meta-models or surrogate models where a
blend of rigorous and rough sub-models is termed hybrid
modeling. Rigorous models are also known as first-
principle models and rough models are empirical models.
The types of models used in planning and scheduling are
mostly rough models where it is common practice to
linearize available rigorous models into first-order Taylor-
series expansions. These linearized models are called base
plus delta, fixed and variable, absolute and relative, slope
with intercept and shift models, (Bodington, 1995).
Data Issues
As is well known in the mathematical programming
community, any decision-making problem can be
decomposed into its model, data and solution. Therefore,
how to collect, clean and compile data for the purposes of
what we call “parameter” feedback merits some discussion.
As mentioned, the focus of this work is to establish a past
rolling horizon (PRH) for planning and scheduling
problems which is symmetrical to the future rolling
horizon (FRH) that exists at the heart of hierarchical
production planning (HPP) (Bitran and Hax, 1977) and
model predictive control (MPC) (Richalet et. al., 1978).
Ideally the data used to perform data reconciliation
and parameter estimation (DRPE), error-in-variables
method (EVM) (Reilly and Patino-Leal, 1981) or
instrumental variables regression (IVR) (Young, 1970)
should be independent and identically normally distributed,
else systemic or gross-errors in the data may exist hence
skewing the results. Unfortunately the data collected after a
plan or schedule has been completed is most often passive
and not perturbed, happenstance and not holistic and
degenerate and not designed. This means that the
calibration or training-data used to fit the key2 model
parameters in the PRH may not be representative of the
production or operating regions or ranges seen in the
control or testing-data found in the FRH. After all, the sole
purpose of planning and scheduling decision-making using
optimization is to push/pull the production to new and
more profitable/performant regimes perhaps not
implemented hitherto. Along this line, the quality of the
data can be classified into three main characteristics: (1)
diversity or richness of the data i.e., all sampled points
span different regions of the control-data, (2) consistency
of the data i.e., all sampled points in both the calibration-
and control-data are taken from the same system and (3)
statistical homogeneity of data i.e., all sampled points in
the calibration- and control-data have the same noise,
error, random shock/perturbation or uncertainty
2 See Krishnan et. al. (1992) or Zyngier (2006) to determine key
model parameters.

3. distributions including their non-linear correlation structure
(Rooney and Biegler, 2001).
To compound the issue, planning and scheduling also
forms a closed-loop feedback control circuit similar to that
found in MPC. The issues with structural analysis and
parameter estimation when closed-loop data is used were
first addressed by Box and MacGregor (1974) when fitting
linear and rational time-series transfer function models.
These issues also exist for planning and scheduling
models. Perhaps one of the main results of their work is to
introduce a small but persistent and uncorrelated dithering
signal or excitation to either the manipulated variables or
the set-points which continuously stimulates the process.
Additionally, closed-loop identification can be
implemented similarly to the approach of Koung and
MacGregor (1993). These same techniques can also be
applied to planning and scheduling optimization systems.
Finally, potential sources of error that exist in the data
arise from several diverse sources as enumerated by Kelly
(2000). They are forecast-errors, measurement-errors,
execution-errors (processing, operating and maintaining),
model structural- and functional-errors (including
decomposition- and aggregation-errors) and last but not
least, solution-errors due to the non-convexity of the
problems (existence of local optima).
Updating and Estimating Methods
The fundamental objective of any model updating and
estimating technique is to find the “best” functional form
which balances the tradeoffs between: (1) the best fit of the
calibration-data i.e., interpolation and (2) the most accurate
and precise parameter estimates when noise exists. There is
a third requirement which is the best prediction of the
control-data i.e., extrapolation, which is the overriding
goal of design-of-experiments and control-relevant
identification. Obviously the quality of the functional form
will depend on both the quality of the structural form and
the quality of the data discussed previously. And, in
advanced process or real-time optimization (RTO)
applications, recognition of the fidelity of the models must
be understood in order to increase the performance of the
models in terms of minimizing the offsets (accuracy)
between the true-plant’s response and the noise-free model
prediction and the variability (precision) of the predictions
due to disturbances (Yip and Marlin, 2004).
Although simple bias-updating is the standard
technique used in both MPC and RTO for “parameter”
feedback, it utilizes a single data point for one time-point
or period in the past and updates the bias, base, intercept or
fixed value of the model formula or equation only from the
measured “variable” feedback. Albeit this is sufficient to
asymptotically remove the offset between the actual and
assumed value of the plant, it is not particularly suited to
planning and scheduling systems. The reason is that in
MPC/RTO, real-time electronic and digitized
measurements of temperatures, pressures, flows, levels,
concentrations and properties are readily available.
Unfortunately, in planning and scheduling applications it
can take days, weeks or months to obtain measured
“variable” feedback given field/control laboratory,
accounting, billing and invoicing system delays.
Instead, a more sophisticated approach is necessary
which continuously collates data over the PRH and
performs a robust method of parameter estimation whilst
respecting errors in the variables. For our purposes, we
choose the method which incorporates simultaneously both
reconciliation and regression from Kelly (1998). This
method is identical to EVM but has useful diagnostics
tailored to the estimability and variability of the both the
reconciled and regressed estimates (Kelly and Zyngier,
2008). More specifically, it reliably computes the
observability of the parameters, the redundancy of the
measurements and the precision of the parameters and
adjusted measurements. In addition, it can also provide
necessary missing-data capability when some data sources
are not available usually intermittently.
Motivating Example
In order to illustrate the importance of “parameter”
feedback in addition to “variable” feedback, a simple
example is presented. In Figure 1, a system is shown that
consists of receiving a supply of material, processing it in a
reactor, storing it in a tank and shipping it to some demand
point. The reactor has a yield of product (Y) which is the
only uncertain parameter in this system. The demands are
exogenously defined by the customer, i.e., it is not a
degree-of-freedom when determining the plan or schedule.
Supply Reactor Tank Demand
True Plant
Tank Holdup
(Variable)
Supply
(Solution)
Reactor Yield
(Parameter)
Demand
(Parameter)
Figure. 1. Closed-Loop System.
At each planning and scheduling cycle, the supply
profile is dispatched to the true plant for implementation
after the solution is calculated. In terms of the feedback
mechanism, two strategies were compared: (i) “variable”
feedback only, i.e., inventory information is available at
the start of each cycle, and (ii) “variable” and “parameter”
feedback, i.e., both inventory and updated yield
information is available. For illustrative purposes, it is
assumed that there is no noise in the measurements and
therefore only regression is necessary to update Ymodel.
The equations used to determine the supply solution at
time-period t (St), given the demand (Dt) and assuming that

4. the inventory in the tank (It) must remain at a constant
target value (Itarget) of 2.0 is provided below.
tY/)DII(S elmodttetargtt  (1)
The inventory It in equation (1) is obtained through
“variable” feedback in that the value of the inventory at the
start of the cycle is measured and is used in the model for
the next cycle. The “true” inventory value is determined
after the supply profile is calculated by using the “true”
plant yield in equation (2):
tDYSII tplantttt   -1 (2)
Initially, Ymodel was assumed to be 0.7 whereas the true
plant yield Yplant has a value of 0.6. The results are shown
in Figure 2 with the demand profile the same for both
scenarios or situations. For the case where the yield is not
updated, i.e., there is “variable” feedback only, the dotted
line inventory profile shows an offset or bias from the
target inventory value of 2.0. By updating the yield at
every cycle using the PRH data, the offset from the
inventory target is quickly corrected (cycle 2) by the time
the Ymodel has been updated to the true value of 0.6.
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8 9
Supply (true) Inventory (true)
Supply (fixed y) Inventory (fixed y)
Supply (updated y) Inventory (updated y)
Figure 2. Supply & Inventory Responses.
Therefore, it is evident that with “variable” feedback
only, it is impossible to remove the persistent offset or
inaccuracy in terms of meeting the planned/scheduled
inventory target of 2.0. Consequently, plan/schedule versus
actual reporting, common place in planning and scheduling
stewardship, will always display a non-zero bias when
significant parameter uncertainty exists of which “variable”
feedback alone will not correct.
Conclusions
Shown in this paper is the limitation of “variable”
feedback when moving from one planning and scheduling
cycle to another. Without both “variable” and “parameter”
feedback, offsets to planning and scheduling targets, set-
points and/or upper/lower bounds will exist similar to the
persistent offset found in proportional-only control and
those observed in real-time process optimization. By
employing reconciliation and regression technology on a
past rolling horizon (PRH) basis, it is possible to reduce
these offsets or inaccuracies asymptotically or evolutionary
over the life-time of the models.
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